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arXiv · 2411.04033

Energy transport in a free Euler-Bernoulli beam in terms of Schr\"odinger's wave function

Serge N. Gavrilov·Anton M. Krivtsov·Ekaterina V. Shishkina
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Abstract

The Schr\"odinger equation is not frequently used in the framework of the classical mechanics, though historically this equation was derived as a simplified equation, which is equivalent to the classical Germain-Lagrange dynamic plate equation. The question concerning the exact meaning of this equivalence is still discussed in modern literature. In this note, we consider the one-dimensional case, where the Germain-Lagrange equation reduces to the Euler-Bernoulli equation, which is used in the classical theory of a beam. We establish a one-to-one correspondence between the set of all solutions (i.e., wave functions $\psi$) of the 1D time-dependent Schr\"odinger equation for a free particle with arbitrary complex valued initial data and the set of ordered pairs of quantities (the beam strain and the particle velocity), which characterize solutions $u$ of the beam equation with arbitrary real valued initial data. Thus, the dynamics of a free infinite Euler-Bernoulli beam can be described by the Schr\"odinger equation for a free particle and vice versa. Finally, we show that for two corresponding solutions $u$ and $\psi$ the mechanical energy density calculated for $u$ propagates in the beam exactly in the same way as the probability density calculated for $\psi$.

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Serge N. Gavrilov, Anton M. Krivtsov, Ekaterina V. Shishkina. 2024-11-06. Energy transport in a free Euler-Bernoulli beam in terms of Schr\"odinger's wave function. https://doi.org/10.1016/j.mechrescom.2025.104382

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